A242805 Integers n such that each of n, n + 1, n + 2, n + 4, n + 5, n + 6 is the squarefree product of three primes.

73293, 120237, 122613, 130429, 143493, 147953, 171893, 180965, 199833, 213153, 219201, 268017, 287493, 298433, 299553, 300093, 313701, 329793, 332889, 341781, 363597, 369393, 376201, 392509, 404453, 432393, 460801, 475809, 493597, 503457, 506517, 508677

Offset: 1

Author

Daniel Constantin Mayer, May 23 2014

Keywords

• nonn

Comments

It is remarkable that this sequence starts with considerably bigger density than the analog A242804 for squarefree integers with two prime divisors. The exceptional density causes the problem that overlapping sextets appear very soon and rather frequently, whereas in A242804 the phenomenon of overlapping sextets does not occur up to the bound 9*10^9.

In fact, there exist 114 nonets n, n + 1, n + 2, n + 4, n + 5, n + 6, n + 8, n + 9, n + 10 of squarefree integers with exactly three prime divisors, up to 10^8. The PARI script in PROG does not start a new sextet before the previous sextet was completed. The impact on bigger clusters, such as nonets and dodekuplets, is illustrated in the CAVEAT of section EXAMPLE.

Examples

			73293 = 3*11*2221, 73294 = 2*13*2819, 73295 = 5*107*137,
73297 = 7*37*283,  73298 = 2*67*547,  73299 = 3*53*461.
CAVEAT:
(1) For the dodekuplet, which starts together with the first nonet,
969833 = 17*89*641,  969834 = 2*3*161639, 969835 = 5*31*6257,
969837 = 3*11*29389, 969838 = 2*173*2803, 969839 = 13*61*1223,
969841 = 23*149*283, 969842 = 2*59*8219,  969843 = 3*7*46183,
969845 = 5*47*4127,  969846 = 2*3*161641, 969847 = 29*53*631.
Not all PARI scripts list 969833 and 969841, but not 969837.
(2) For the second nonet,
1450257 = 3*229*2111, 1450258 = 2*179*4051, 1450259 = 83*101*173,
1450261 = 29*43*1163, 1450262 = 2*11*65921, 1450263 = 3*191*2531,
1450265 = 5*23*12611, 1450266 = 2*3*241711, 1450267 = 7*13*15937,
the PARI script lists 1450257 only, but not 1450261.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 176 terms from Zak Seidov)

Crossrefs

Cf. A242793 and A242804 (two primes), A242806 (four primes), A242829 (five primes).

Programs

Mathematica

s = {}; Do[If[AllTrue[{k, k + 1, k + 2, k + 4, k + 5, k + 6}, SquareFreeQ] && {3, 3, 3, 3, 3, 3} == PrimeOmega[{k, k + 1, k + 2, k + 4, k + 5, k + 6}], AppendTo[s, k]], {k, 73293, 2000000, 4}]; s (* Zak Seidov, Nov 12 2018 *)

PARI

{ default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=3; for(n=lb, ub, if(issquarefree(n)&&(o==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1
				

PARI

is(n) = {my(f=factor(n)); matsize(f)==[3, 2] && vecmax(f[ , 2])==1};
isok(v) = vecextract(v, "^4")==[1, 1, 1, 1, 1, 1]; v = vector(7); for(k=8, 550000, v=concat(vecextract(v, "^1"), is(k+6)); if(isok(v), print1(k, ", "))) \\ Amiram Eldar, Nov 13 2018

PARI

upto(n) = {my(res = List(), streak = 1); for(i = 31, n+6, if(factor(i)[, 2] == [1, 1, 1]~, streak++; if(streak % 4 == 3 && streak >= 7, listput(res, i-6)), if(streak % 4 == 3, streak++, streak = 0))); res} \\ David A. Corneth, Nov 13 2018